Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.

On some general scheme of constructing iterative methods for searching the Nash equilibrium in concave games

The subject of the paper is finite-dimensional concave games id est noncooperative $n$-person games with objective functionals concave with respect to `their own' variables. For such games we investigate the problem of designing iterative algorithms for searching the Nash equilibrium with convergence guaranteed without requirements concerning objective functionals such as smoothness and as convexity in `strange' variables or another similar hypotheses (in the sense of weak convexity, quasiconvexity and so on). In fact, we prove some assertion analogous to the theorem on convergence of $M$-Fej\'{er iterative process for the case when an operator acts in a finite-dimensional compact and nearness to an objective set is measured with the help of arbitrary continuous function. Then, on the base of this assertion we generalize and develop the approach suggested by the author formerly to searching the Nash equilibrium in concave games. The last one can be regarded as "a cross between" the relaxation algorithm and the Hooke-Jeeves method of configurations (but taking into account a specific character of the the residual function being minimized). Moreover, we present results of numerical experiments with their discussion.

Case Study: Mapping Evoked Fields in Primary Motor and Sensory Areas via Magnetoencephalography in Tetraplegia

Devices interfacing with the brain through implantation in cortical or subcortical structures have great potential for restoration and rehabilitation in patients with sensory or motor dysfunction. Typical implantation surgeries are planned based on maps of brain activity generated from intact function. However, mapping brain activity for planning implantation surgeries is challenging in the target population due to abnormal residual function and, increasingly often, existing MRI-incompatible implanted hardware. Here, we present methods and results for mapping impaired somatosensory and motor function in an individual with paralysis and an existing brain–computer interface (BCI) device. Magnetoencephalography (MEG) was used to directly map the neural activity evoked during transcutaneous electrical stimulation and attempted movement of the impaired hand. Evoked fields were found to align with the expected anatomy and somatotopic organization. This approach may be valuable for guiding implants in other applications, such as cortical stimulation for pain and to improve implant targeting to help reduce the craniotomy size.

Leber’s Congenital Amaurosis: Current Concepts of Genotype-Phenotype Correlations

Leber’s congenital amaurosis (LCA), one of the most severe inherited retinal dystrophies, is typically associated with extremely early onset of visual loss, nystagmus, and amaurotic pupils, and is responsible for 20% of childhood blindness. With advances in molecular diagnostic technology, the knowledge about the genetic background of LCA has expanded widely, while disease-causing variants have been identified in 38 genes. Different pathogenetic mechanisms have been found among these varieties of genetic mutations, all of which result in the dysfunction or absence of their encoded proteins participating in the visual cycle. Hence, the clinical phenotypes also exhibit extensive heterogenicity, including the course of visual impairment, involvement of the macular area, alteration in retinal structure, and residual function of the diseased photoreceptor. By reviewing the clinical course, fundoscopic images, optical coherent tomography examination, and electroretinogram, genotype-phenotype correlations could be established for common genetic mutations in LCA, which would benefit the timing of the diagnosis and thus promote early intervention. Gene therapy is promising in the management of LCA, while several clinical trials are ongoing and preliminary success has been announced, focusing on RPE65 and other common disease-causing genes. This review provides an update on the genetics, clinical examination findings, and genotype-phenotype correlations in the most well-established causative genetic mutations of LCA.

Gage MPC: Bypassing Residual Function Leakage for Non-Interactive MPC

Existing models for non-interactive MPC cannot provide full privacy for inputs, because they inherently leak the residual function (i.e., the output of the function on the honest parties’ input together with all possible values of the adversarial inputs). For example, in any non-interactive sealed-bid auction, the last bidder can figure out what was the highest previous bid. We present a new MPC model which avoids this privacy leak. To achieve this, we utilize a blockchain in a novel way, incorporating smart contracts and arbitrary parties that can be incentivized to perform computation (“bounty hunters,” akin to miners). Security is maintained under a monetary assumption about the parties: an honest party can temporarily supply a recoverable collateral of value higher than the computational cost an adversary can expend. We thus construct non-interactive MPC protocols with strong security guarantees (full security, no residual leakage) in the short term. Over time, as the adversary can invest more and more computational resources, the security guarantee decays. Thus, our model, which we call Gage MPC, is suitable for secure computation with limited-time secrecy, such as auctions. A key ingredient in our protocols is a primitive we call “Gage Time Capsules” (GaTC): a time capsule that allows a party to commit to a value that others are able to reveal but only at a designated computational cost. A GaTC allows a party to commit to a value together with a monetary collateral. If the original party properly opens the GaTC, it can recover the collateral. Otherwise, the collateral is used to incentivize bounty hunters to open the GaTC. This primitive is used to ensure completion of Gage MPC protocols on the desired inputs. As a requisite tool (of independent interest), we present a generalization of garbled circuit that are more robust: they can tolerate exposure of extra input labels. This is in contrast to Yao’s garbled circuits, whose secrecy breaks down if even a single extra label is exposed. Finally, we present a proof-of-concept implementation of a special case of our construction, yielding an auction functionality over an Ethereum-like blockchain.

Least Square Methods Based on Control Points of Said Ball Curves for Solving Ordinary Differential Equations

This paper presents the use of Said Ball curve’s control points to approximate the solutions of linear ordinary differential equations (ODEs). Least squares methods (LSM) is proposed to find the control points of Said Ball curves by minimizing the error of residual function.The residual error is measured by taking the sum of squares of the Said Ball curve’s control points of the residual function. Then the approximate solution of ODEs is obtained by minimizing residual error.Two numerical examples are given in term of error and compared with the exact solution to demonstrate the efficiency of the proposed method.